Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion

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Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion

Victor Péron, Gabriel Caloz, Monique Dauge

To cite this version:

Victor Péron, Gabriel Caloz, Monique Dauge. Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion. WAVES 2009 The 9th International Conference on Mathematical and Numerical Aspects of Waves Propagation, Jun 2009, Pau, France. �inria-00528514�

Skin-Effect Description in Electromagnetism with a Scaled Asymptotic Expansion V. P´eron^1,∗, G. Caloz¹, M. Dauge¹

1 Numerical Analysis, IRMAR, University of Rennes 1, Rennes, France

∗Email: victor.peron@univ-rennes1.fr Abstract

We study a transmission problem in high contrast media. The 3-D case of the Maxwell equations in harmonic regime is considered. We derive an asymp- totic expansion with respect to a small parameter δ >0 related to high conductivity. This expansion is theoretically justified at any order. Numerical sim- ulations highlight the skin-effect and the expansion accuracy.

Introduction

We consider the diffraction problem of waves by highly conducting materials in electromagnetism.

The high conductivity reduces the penetration of the wave to a boundary layer, see [1]. The physical model is the following. Ωcd is an open bounded domain in R³ with connected complement, occupied by a con- ducting medium. Ω^cd is embedded in an insulating medium Ω^is. We suppose that their common inter- face Σ is smooth. We define Ω = Ωcd ∪Σ∪Ωis. We denote by δ a small parameter which is inversely proportional to the square root of the conductivity σ. The depth of the boundary layer is proportional to δ. We first give the formal construction of the asymptotic expansion. Then, we prove optimal error estimates. Finally, we present numerical simulations in axisymmetric geometry.

1 Scaled asymptotic expansion

Eliminating the magnetic field H_δ from Maxwell equations, we perform a study in electric fieldE_δ. 1.1 Normal coordinates

To describe the boundary layer in Ω^cd, we define a local normal coordinate system (yα, y₃), α∈ {1,2}, in a tubular neighborhood O of Σ, y₃ ∈(0, h₀). The euclidian metric inOis denoted by (g_ij)_{i,j∈{1,2,3}}. We adopt the tensorial calculus, see [2], to write Maxwell equations in these coordinates. The curl operator writes

( (curlE)^α = √¹g^3βα(∂₃E_β−∂_βE₃) (curlE)³ = √¹g^3αβD_α^hE_β

with g = det(gij), ^ijk the Levi-Civita symbol, and D^h a covariant derivative defined for h=y3, see [2].

1.2 Scaling and ansatz

We perform the scaling Y₃= ^y_δ³ inO, and expand the 3D Maxwell operator in power of series ofδ. This leads to postulate the following expansions

E^is_δ(x)∼X

j≥0

E^is_j(x)δ^j, E^cd_δ (x)∼X

j≥0

E^cd_j (x;δ)δ^j (1)

respectively in Ω^is, and Ω^cd, with E^cd_j (x;δ) = W^cd_j (yα, ^y_δ³) when (yα, y3) ∈ O. We prove that W^cd₀ = 0, and W^cd_j (., Y₃) = exp(−λY₃)[a₀ +.. + aj−1Y₃^j−1], with <(λ) > 0. According to Faraday’s law H_δ = 1/(iκµ₀) curlE_δ, κ > 0 (a wave number), we derive the expansion of the magnetic field.

2 Uniform a priori estimate

A regularized variational formulation of our prob- lem in the domain Ω using the space X_N(δ) = {E ∈ H(curl,Ω)| ε(δ)E ∈ H(div,Ω), E × n = 0 on ∂Ω} is:

Find E^δ ∈X_N(δ) such that for all u∈X_N(δ) Z

Ω

curlE·curlu+divε(δ)Edivε(δ)u−κ²ε(δ)E·udx

= Z

Ω

(F^δ·u− 1

κ² divF^δdivε(δ)u)dx, (2) with ε(δ) = _µ¹

0(1_Ωis + (1 +_δⁱ2)1_Ωcd), µ₀ > 0, and F^δ ∈ H(div,Ω). Under a spectral hypothesis on κ, we prove that∃δ₀ >0 small enough such that ∀δ ∈ (0, δ₀), (2) has a unique solution E^δ ∈ X_N(δ), and there is a constantC >0 independent ofδ such that

kcurlE^δk0,Ω+kdivε(δ)E^δk0,Ω+kE^δk0,Ω

+1

δkE^δk0,Ωcd ≤CkF^δkH(div,Ω). We prove this estimate using a technique of vector potential, see [3]. Defining R^δ_m from (1) by remov- ing to E_δ them+ 1 first-terms of the expansion, we deduce optimal error estimates for the truncated ex- pansions

kR^δ_mkH(curl,Ω)+kdivε(δ)R^δ_mk0,Ω+1

δkR^δ_mk0,Ωcd ≤Cδ^m−1.

3 Numerical simulations

Here we present numerical experiments to illus- trate the accuracy of the asymptotic expansion. We perform our analysis to the magnetic field and re- strict ourselves to axisymmetric domains with an or- thoradial data. After having meshed the domain, we compute the orthoradial component of the mag- netic field Hθ with the finite element library Melina, see [4]. The numerical method uses a variational form with unknown Hθ(r, z) in a meridian domain, which is meshed in such a way the boundary layer is correctly taken into account, see Figure 1 for spheroidal domains. We extract values of |Hθ(y3)|

Figure 1: |=Hθ| forσ= 5S.m⁻¹ and 80S.m⁻¹

along edges of the mesh for z = 0. We perform a linear regression from values of log₁₀|H_θ(y₃)| in a depth-skin defined by skin(σ) = √

2δ(σ)/κ, cor- responding to n(σ) points on the mesh, see Fig- ure 2 for spherical domains with y₃ = 2−r. We

0 0.5 1 1.5 2

−6

−5

−4

−3

−2

−1 0 1 2

2 − r

Figure 2: log₁₀|Hθ(2−r)| forσ= 5S.m⁻¹

get a ”numerical” slope s(σ). Let H^cd_θ,1 be the or- thoradial component of H^cd₀ (x;δ) + δH^cd₁ (x;δ) (the truncated asymptotic expansion of Hδ). We prove that log₁₀|H_θ,1^cd(y₃)|=α(σ)y₃+ log₁₀|H_θ,0^is

|Σ|+O(δ).

α(σ) = β−1/skin(σ)

/ln 10 is a ”theoretical” slope, andβ depends on the curvature on Σ. The accuracy of the expansion is tested by representing the relative error between ”numerical” and ”theoretical” slopes:

error(σ) =|α(σ)−s(σ)|

|α(σ)|, see Figure 3.

σ (S.m⁻¹) 5 20 80

skin(σ) (cm) 10.3 5.15 2.58

n(σ) 7 5 3

s(σ) −3,65542 −7,88390 −16,27916 α(σ) −3,67331 −7,88950 −16,32188 error(σ) (%) 0,48 0,07 0,26

Figure 3: Relative error in spheroidal domains

References

[1] H. Haddar, P. Joly, H.N. Nguyen, Construction and analysis of approximate models for electro- magnetic scattering from imperfectly conducting scatterers, Math. Models Methods Appl. Sci., vol. 18, 10, 1787-1827, 2008.

[2] E. Faou,Elasticity on a thin shell: Formal series solution, Asymptotic Analysis,31(3-4):317-361, 2002.

[3] C. Amrouche, C. Bernardi, M. Dauge, V.Girault, Vector Potentials in Three- dimensional Non-smooth Domains, Math.

Meth. Appl. Sci., 21, 823-864, 1998.

[4] D. Martin, Melina. http://anum-maths.univ- rennes1.fr/melina/